For \(n=100\)  the behavior is even more interesting: \(\Delta m^-(t)\)  reaches a negative minimum at short times, then increases to become positive, reaches a positive maximum at large times and then starts decreasing strongly again to become negative. As we explain below, this rich behavior is a result of interplay between the sensing and adaptation modules of the network [1]}. For large receptor clusters, the activity fluctuation in the cell is quite strong. When the activity becomes too high or too low, to restore it to its average level, adaptation needs to play an important role and can sometimes override the signal coming from ligand concentration variation in the cell's environment. For small \(t\) , the behavior of \(\Delta m^ -(t)\)  is controlled by high activity runs. This can be clearly seen in Fig. REF (f) where number of high activity runs at short times is much higher than low activity ones. Note that Fig. REF  show data for uphill runs but we find very similar variation for the downhill runs as well. High activity runs cause strong demethylation . However, this demethylation process lowers the activity significantly and even though ligand concentration is dropping rapidly, it cannot keep up with the strong demethylation. The resulting low activity triggers methylation (see our data in Fig. REF (c) where even for high activity downhill runs the initial demethylation slows down with time). Therefore after reaching a minimum, \(\Delta m^ -(t)\)  starts increasing again and even changes sign to become positive. At large times, when only very long runs survive in \(N^-(t)\)  population, the drop in ligand free energy along such long downhill trajectories becomes quite large which can now compete against the strong methylation variation experienced by the receptors. Therefore, the activity starts increasing again which explains the maximum of \(\Delta m^ -(t)\)  followed by a drop (our data for low activity downhill runs in Fig. REF (c) show this trend clearly). We have presented additional supporting data in Appendix .
